For the purposes of understanding the present disclosure, it is useful to consider a representation of the total optical E-field E(t) as a vector confined to a plane and emanating from a fixed origin, where the length of the vector gives the amplitude of the E-field at any instant (t), and the direction of the vector gives the phase of the field at any instant (t). Within this construction, we consider two basis sets. The first basis set is a Cartesian coordinate system centered on the E-field origin. In this Cartesian representation, the total E-field E(t) is decomposed along the orthogonal Real (Re) and Imaginary (Im), or, equivalently, In-phase (I) and Quadrature (Q), directions. The second basis set is a polar coordinate system, again sharing its origin with that of the E-field vector. In this polar representation, the E-field is decomposed into vector length (S) and phase angle (φ) relative to the Re-direction. These two basis sets are related by a non-linear transformation, in a manner well known in the art. In each of these representations, the time-sequence of loci of the end-point of the E-field vector may be referred to as a trajectory of the E-field.
The present disclosure discusses modulation formats and signals in which the state (e.g. amplitude and/or phase) of the signal at any instant depends on the state of the signal both before and after that instant. For example, the present disclosure discusses a Constrained Continuous Phase Modulation (C-CPM) scheme, in which the state of the modulated signal corresponding to a given symbol depends not only on the value of that symbol, but also on the values of the symbols that precede and follow it. Modulation formats and signals that display this characteristic may be referred to as having “memory”.
In the optical communications space, various techniques are used to synthesize an optical communications signal for transmission. FIG. 1 illustrates a dual-polarization transmitter known in the art, which is designed to transmit two data signals on respective orthogonal polarizations of an optical wavelength channel. The laser 2 generates a narrow-band continuous wave (CW) optical carrier 4 having a desired wavelength. A beam splitter (or power divider) 6 separates the carrier 4 into a pair of linearly polarized CW lights, which are supplied to a respective E/O converter 8 which operates to modulate the amplitude and/or phase of the CW light to generate a respective polarization signal 10 based on one or more drive signals S(t) generated by a driver circuit 12 based on a respective input data signal x(t) and y(t). The two polarizations signals are then combined using a polarization combiner 14 to yield a polarization multiplexed optical communications signal 16 for transmission to a receiver.
In the arrangement illustrated in FIG. 1, each E/O converter 8 is provided a nested Mach-Zehnder (MZ) modulator known in the art. This arrangement enables the transmitter to utilize a variety of modulation and encoding schemes, to obtain high spectral efficiency and data transmission speeds. For example, transmitters using optical coherent modulation formats such as dual polarization multiplexed Quadrature Phase Shift Keying (DP-QPSK) and a nested Mach Zehnder E/O modulators constructed using Lithium Niobate (or Indium Phosphide) have recently enabled transmission rates exceeding 40 Gbps (Giga-bits-per-second) per optical wavelength channel over geographical distances of several hundred kilometers. A limitation of these transmitters, however, is that the optical components, particularly the E/O converter 8, are expensive.
Techniques that enable high speed communications with low-cost optical components remain highly desirable.